# Thread: Imaginary Number raised to an Imaginary Number??

1. I got a question wrong on an Exam in Applied Electromagnetics (An engineering course)...

True or False-
j^(j) has infinitely many values, all of which are real numbers...?

Why?  2.

3. So what does i<sup>i</sup> mean? (I'm a mathematician, so I can't help but use i, sorry.) It means e<sup>i ln i</sup>. Okay, so what does ln i mean? It's any complex number z such that e<sup>z</sup> = i, and we can calculate it as ln i = ln|i| + i arg i, where arg i is the angle that i makes with the positive real axis. ln|i| = ln 1 = 0, and arg i = π/2... or 5π/2 or 9π/2 or any number of the form (4n+1)π/2, where n is any integer. So ln i = i(4n+1)π/2 for any integer, and thus:

i<sup>i</sup> = e<sup>i*i(4n+1)π/2</sup> = e<sup>-(4n+1)π/2</sup>

Neat.  4. Very Nice. Makes perfect sense. Problem sure did get me thinkin...

Thank you much.  Bookmarks
 Posting Permissions
 You may not post new threads You may not post replies You may not post attachments You may not edit your posts   BB code is On Smilies are On [IMG] code is On [VIDEO] code is On HTML code is Off Trackbacks are Off Pingbacks are Off Refbacks are On Terms of Use Agreement